**1. Introduction**

Many liquids such as detergents, printer ink, animal blood, foodstuff, paints, polymer fluids, etc., transform their properties of flow subjected to operating shear stress, and thus diverge from viscous fluids. These fluids are identified as non-Newtonian substances. Numerous researchers have reported different non-Newtonian fluid models and a few of them are micropolar, Casson, Burgers, Sisko, Maxwell, Oldroyd-B, generalized Burgers, and Cross models, etc. In this paper, we report the Cross liquid [1] model, which states features of stress. In addition, this model sufficiently distinguishes the flow in the region of the power law and high, as well as low, regions of shear rates. In this study, unlike

the fluid of power law, first, we achieve a finite viscosity as the rate of shear disappears which also involves a time constant owing to the importance of this model in numerous industrial and engineering computations. Utilization of Cross fluid in industries comprises the polymer latex of the aqueous solution and blood, as well as solutions of synthesis polymeric. Khan et al. [2] inspected the flow of Cross liquid through heat transfer from a planar stretched sheet and found the numerical solution through the shooting technique. The impact of electric field with the characteristic of heat transfer involving Cross liquid from a stretched sheet was scrutinized by Hayat et al. [3] who found that the liquid velocity grew with a rising Weissenberg parameter while temperature distribution decayed due to the Pr. Khan et al. [4] scrutinized the axisymmetric flow and the characteristic of heat transfer containing Cross liquid using a radial stretched sheet and observed that the power-law index raised the structure of the velocity boundary layer. Ijaz Khan et al. [5] scanned the activation energy impact on the magnetic flow of Cross liquid from a stretched surface. Another study, by Ijaz Khan et al. [6], surveyed the magnetic influence on mixed convective flow involving Cross nanofluid with activation energy. Recently, Azam et al. [7] applied the concept of solar energy on time-dependent flow in the presence of Cross nanofluid from a stretched sheet with nonlinear radiation.

The impact of non-Newtonian liquids in the porous media is significant in the fields of engineering and industries due to its numerous applications such as mud injections, cement or slurry grouts to strengthen soils, blood circulation through the kidney, insulation of fibrous, electrochemistry, and drilling liquid injection in rocks for ornamental oil recovery, or for the fortification of the well, etc. Bejan et al. [8], Vafai [9], and Vadasz [10] discussed further applications in their books. Darcy's law has been utilized generally to inspect the behavior of flow in a porous medium. However, the connection between the velocity of flow and pressure gradient at rates of high flow cannot be modeled through Darcy's law (Spivey et al. [11]). There is further indication that at a high rate of flow, the non-Darcy involve several subsurface systems of biological porous and engineering porous flow [12–14]. Forchheimer [15] included a term of velocity squared in the Darcy to analyze the boundary and inertia aspects. This term is constantly applied to larger Reynolds numbers. Rashidi et al. [16] discovered the influence of electric field on fluid flow with the characteristic of heat transfer in a Darcy–Brinkman–Forchheimer medium. The impact of variable thermal conductivity of Darcy–Forchheimer flow in the presence of Cattaneo–Christov heat-flux was considered by Hayat et al. [17]. In another paper, Hayat et al. [18] examined the non-Newtonian viscoelastic fluid involving nanoliquid through nonlinear stretched surface engrossed in the Darcy–Forchheimer porous medium. Kang et al. [19] employed finite di fference technique to discuss the Neumann condition for the general Darcy–Forchheimer problem. Hayat et al. [20] explored the homogenous-heterogeneous reaction of viscous liquid in a Darcy–Forchheimer porous medium through a curved stretched surface. They scrutinized that the porosity and inertia parameters produce larger temperature. Recently, Rasool [21] considered the Darcy–Forchheimer flow to investigate electric field containing nanoparticle through a nonlinear stretched surface. They observed that the mass and heat flux decline due to porosity while drag force is enhanced. A few other similar studies are given in [22–24].

As mentioned above, the present literature is packed with works comprising the heat transfer characteristics of boundary-layer flow involving Newtonian and non-Newtonian liquids. In addition, the research regarding the Darcy–Forchheimer flow through heat transport comprising Cross liquid has disclosed a vital pledge in industrial and environmental systems, such as the process of fermentation, petroleum resources, usage of geothermal energy, production of crude oil, grain storage, etc. However, the review of literature revealed that no one has considered the impact of slip e ffects on mixed convection flow of Cross liquid in the porous media. Therefore, in this research, we focus our attention to the Darcy–Forchheimer flows involving non-Newtonian Cross liquids from a vertical plate with mixed convection and slip e ffects. Similarity variables are employed to metamorphose the PDEs into nonlinear ODE's. The metamorphosed system is then exercised through bvp4c solver. The dual nature of solutions is acquired in opposing flow. The vital constraints in the flow field are discussed via graphical portraits.
